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Thread: Imaginary Number raised to an Imaginary Number??

  1. #1 Imaginary Number raised to an Imaginary Number?? 
    Forum Freshman
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    Apr 2008
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    I got a question wrong on an Exam in Applied Electromagnetics (An engineering course)...

    True or False-
    j^(j) has infinitely many values, all of which are real numbers...?

    answer is True!

    Why?


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  3. #2  
    Forum Professor serpicojr's Avatar
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    So what does i<sup>i</sup> mean? (I'm a mathematician, so I can't help but use i, sorry.) It means e<sup>i ln i</sup>. Okay, so what does ln i mean? It's any complex number z such that e<sup>z</sup> = i, and we can calculate it as ln i = ln|i| + i arg i, where arg i is the angle that i makes with the positive real axis. ln|i| = ln 1 = 0, and arg i = π/2... or 5π/2 or 9π/2 or any number of the form (4n+1)π/2, where n is any integer. So ln i = i(4n+1)π/2 for any integer, and thus:

    i<sup>i</sup> = e<sup>i*i(4n+1)π/2</sup> = e<sup>-(4n+1)π/2</sup>

    Neat.


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  4. #3  
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    Very Nice. Makes perfect sense. Problem sure did get me thinkin...

    Thank you much.
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